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Properties

Label 4-17056-1.1-c1e2-0-1
Degree $4$
Conductor $17056$
Sign $-1$
Analytic cond. $1.08750$
Root an. cond. $1.02119$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 2·5-s − 8-s − 4·9-s + 2·10-s − 5·13-s + 16-s + 2·17-s + 4·18-s − 2·20-s − 6·25-s + 5·26-s − 32-s − 2·34-s − 4·36-s − 6·37-s + 2·40-s − 11·41-s + 8·45-s + 6·49-s + 6·50-s − 5·52-s − 8·61-s + 64-s + 10·65-s + 2·68-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.353·8-s − 4/3·9-s + 0.632·10-s − 1.38·13-s + 1/4·16-s + 0.485·17-s + 0.942·18-s − 0.447·20-s − 6/5·25-s + 0.980·26-s − 0.176·32-s − 0.342·34-s − 2/3·36-s − 0.986·37-s + 0.316·40-s − 1.71·41-s + 1.19·45-s + 6/7·49-s + 0.848·50-s − 0.693·52-s − 1.02·61-s + 1/8·64-s + 1.24·65-s + 0.242·68-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(17056\)    =    \(2^{5} \cdot 13 \cdot 41\)
Sign: $-1$
Analytic conductor: \(1.08750\)
Root analytic conductor: \(1.02119\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 17056,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2$C_1$ \( 1 + T \)
13$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 4 T + p T^{2} ) \)
41$C_1$$\times$$C_2$ \( ( 1 + T )( 1 + 10 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \) 2.3.a_e
5$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.5.c_k
7$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \) 2.7.a_ag
11$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \) 2.11.a_u
17$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.17.ac_ba
19$C_2^2$ \( 1 + 28 T^{2} + p^{2} T^{4} \) 2.19.a_bc
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.23.a_abe
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.29.a_w
31$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \) 2.31.a_abe
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.37.g_cg
43$C_2^2$ \( 1 - 62 T^{2} + p^{2} T^{4} \) 2.43.a_ack
47$C_2^2$ \( 1 - 34 T^{2} + p^{2} T^{4} \) 2.47.a_abi
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) 2.53.a_dy
59$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \) 2.59.a_dm
61$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) 2.61.i_fe
67$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \) 2.67.a_abs
71$C_2^2$ \( 1 - 86 T^{2} + p^{2} T^{4} \) 2.71.a_adi
73$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.73.ae_by
79$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \) 2.79.a_bu
83$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \) 2.83.a_ek
89$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) 2.89.aba_na
97$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) 2.97.g_gw
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.67255191569615668978704337226, −10.18820359399451393619231876152, −9.691646095150103502629476239958, −8.976658360878331663565062124421, −8.599801419066007433963583712326, −7.80738954231211427157540197942, −7.69647464561071274178418598725, −6.93958469967566024181537829470, −6.20430128238977017264053445043, −5.46279112591124427155203688746, −4.88636263098433199433415548971, −3.79230967290255456478462215822, −3.10046576475329527166864456407, −2.13013716394441461426996146730, 0, 2.13013716394441461426996146730, 3.10046576475329527166864456407, 3.79230967290255456478462215822, 4.88636263098433199433415548971, 5.46279112591124427155203688746, 6.20430128238977017264053445043, 6.93958469967566024181537829470, 7.69647464561071274178418598725, 7.80738954231211427157540197942, 8.599801419066007433963583712326, 8.976658360878331663565062124421, 9.691646095150103502629476239958, 10.18820359399451393619231876152, 10.67255191569615668978704337226

Graph of the $Z$-function along the critical line